Resolutions of two conjectures on the spectral diameter

Let $λ_1(G) \geq \dots \geq λ_n(G)$ be the adjacency spectrum of a graph $G$ on $n$ vertices. The spectral distance $σ(G,H)$ between $n$-vertex graphs $G$ and $H$ is the Manhattan distance between their spectra, i.e. $σ(G,H) = \sum_{i=1}^n |λ_i(G) - λ_i(H)|$. Given a set $\mathcal{G}$ of pairwise no